Arrayed waveguide grating with waveguides of unequal widths

ABSTRACT

The present invention provides an arrayed waveguide such that each waveguide of the grating has a substantially uniform width, but the width of any single waveguide in the grating is selected based on a predetermined birefringence required for the waveguide. Generally, the narrowest grating waveguide has the longest overall length and the widest grating waveguide has the shortest overall length. The remaining intermediate waveguides have widths that are interpolated between the narrowest and widest waveguide gratings. With an appropriate width for each waveguide, an arrayed waveguide grating is provided that has low polarization dependent wavelength.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a conversion and claims the benefit of priority toU.S. Provisional Patent Application Ser. No. 60/276,779, filed on Mar.16, 2001, which is incorporated herein by reference in its entirety.

FIELD OF THE INVENTION

This invention relates generally to planar lightwave circuits for use inoptical signal routing applications, in particular, planar lightwavecircuits having arrayed waveguide gratings.

BACKGROUND OF THE INVENTION

The increase in Internet traffic and other telecommunications over thepast several years has caused researchers to explore new ways toincrease fiber optic network capacity by carrying multiple data signalsconcurrently through telecommunications lines. To expand fiber networkcapacity, fairly complex optical components have already been developedfor wavelength division multiplexing (WDM) and dense wavelength divisionmultiplexing (DWDM).

In a WDM system, multiple optical data signals of different wavelengthsare added together in a device called a multiplexer and the resultingdata signal is transmitted over a fiber optic cable. The wavelengthdivision multiplexed signal comprises a plurality of optical signalshaving a predetermined nominal wavelength difference from each other. Ademultiplexer separates the multiple optical data signals of differentwavelength. Any WDM system must include at least one component toperform the function of optical multiplexing (namely, the multiplexer)and at least one component to perform the function of opticaldemultiplexing (namely, the demultiplexer). The optical multiplexer andthe optical demultiplexer are each examples of optical wavelengthrouters.

In general, an optical wavelength router has at least one input opticalport and at least one output optical port. In an optical router, lightmay be transmitted from a specific input port to a specific output portonly if the light has an appropriate wavelength. Complex WDM systems mayrequire optical wavelength router components that are more complex thana multiplexer or a demultiplexer.

Planar lightwave circuit technology is one technology that may be usedto implement an optical wavelength router. A planar lightwave circuit(PLC) is an application of integrated optics. In a PLC, light isrestricted to propagate in a region that is thin (typically betweenapproximately 1 μm and 30 μm) in one dimension, referred to herein asthe lateral dimension, and extended (typically between 1 mm and 100 mm)in the other two dimensions. The plane in which the PLC is disposed isdefined as the plane of the PLC. The longitudinal direction is definedas the direction of propagation of light at any point on the PLC. Thelateral direction is defined to be perpendicular to the plane of thePLC. The transverse direction is defined to be perpendicular to both thelongitudinal and the lateral directions.

In a typical example of a PLC, a slab waveguide comprises three layersof silica glass, a core layer lying between a top cladding layer and abottom cladding layer. Channel waveguides are often formed by at leastpartially removing (typically with an etching process) core materialbeyond the transverse limits of the channel waveguide and replacing itwith at least one layer of side cladding material that has an index ofrefraction that is lower than that of the core material. The sidecladding material is usually the same material as the top claddingmaterial. In this example, each layer is doped in a manner such that thecore layer has a higher index of refraction than either the top claddingor bottom cladding. When layers of silica glass are used for the opticallayers, the layers are typically deposited on a silicon wafer.Deposition processes may include, chemical vapor deposition (CVD), lowpressure chemical vapor deposition (LPCVD), and/or plasma-enhanced CVD(PECVD). As a second example, slab waveguides and channel waveguidescomprise three or more layers of InGaAsP. In this example, adjacentlayers have compositions with different percentages of the constituentelements In, P, Ga, and As. As a third example, one or more of theoptical layers of the slab waveguide and/or channel waveguide maycomprise an optically transparent polymer. In this example, spin coatingis one known film deposition method. Another example of a slab waveguidecomprises a layer with a graded index such that the region of highestindex of refraction is bounded by regions of lower indices ofrefraction. Graded index structures are commonly formed by dopantin-diffusion and have been used for LiNbO₃ waveguides. A doped-silicawaveguide is usually preferred because it has a number of attractiveproperties including low cost, low loss, low birefringence, stability,and compatibility for coupling to fiber.

The arrayed-waveguide grating router (AWGR) is the preferred integratedoptical router. An AWGR is a planar lightwave circuit comprising atleast one input channel waveguide, an input planar waveguide, anarrayed-waveguide grating (AWG), an output planar waveguide, and atleast one output channel waveguide. The edge of the input planarwaveguide to which the input channel waveguides are attached is referredto herein as the input focal curve. The edge of the output planarwaveguide to which the output channel waveguides are attached isreferred to herein as the output focal curve. The arrayed-waveguidegrating comprises an array of channel waveguides. The length of thei^(th) waveguide in the AWG is denoted as L_(i). The angular dispersionthat is provided by the AWG is determined in part by the difference inlength between adjacent waveguides, L_(i+1)−L_(i). The details ofconstruction and operation of the AWGR are described in K. Okamoto,Fundamentals of Optical Waveguides, pp. 346-381, Academic Press, SanDiego, Calif., USA (2000). Each of the publications and patents referredto in this application are herein incorporated by reference in theirentirety.

FIG. 1A depicts a conventional AWG router (AWGR) that acts as ademultiplexer 10. A plurality of optical signals incident on one inputoptical port propagates through the device in the following sequence:the signals propagate through an input waveguide 12, which is a inputwaveguide associated with the input port; through an input slabwaveguide 14, which has the function of expanding the optical field inthe transverse direction by diffraction; through the dispersive region16 (namely, the array waveguide region) comprising an array of AWGwaveguides 18 for modifying the direction of propagation for eachwavelength constituent according to the wavelength of the constituent ofthe plurality of signals; through an output slab waveguide 20 forfocusing the signals of different wavelength coupled from the dispersiveregion 16 into a plurality of predetermined positions in accordance withthe predetermined wavelength difference; through a plurality of outputwaveguides 22 each associated with one output port. FIG. 1A depicts anAWG comprising six waveguide; however, any number of waveguides may beused and herein the number of waveguides used is referred to as “N.” Arepresentative cross-section 30, section 1B-1B, of waveguide gratings 16from FIG. 1A is shown in FIG. 11B. Depicted are the substrate 34, thebottom cladding 36, the top cladding 38, and waveguides comprising corematerial 31, 32, 33. These waveguides are typically buried channelwaveguides as shown and typically have a core region with uniform heightand width as seen in first, intermediate, and N^(th) waveguides 31, 32,33, respectively. That is, the height of each waveguide of the gratingis identical and the width of each waveguide of the grating isidentical.

The dispersive property of the arrayed waveguide grating (AWG) region isattributable to the construction of the plurality of waveguides withinthe waveguide grating region such that adjacent waveguides have apredetermined length difference in accordance to the required dispersiveproperties of the dispersive region 16, so that each signal at differentwavelength coupled to and traveling over each channel waveguide 18 isprovided with a phase difference from each other in accordance with thepredetermined length difference. Each of the output waveguides 22includes an input end 24, which is arranged at a predetermined position,so that each separated signal at each wavelength is coupled to eachoutput waveguide 22 and emerges from an output end 26 thereof.

In operation, the wavelength division multiplexed signals coupled intothe input channel waveguide 12 expand into the input slab waveguide 14by diffraction. Then, the expanded signals are distributed to thechannel waveguides 18 of the arrayed-waveguide grating 16. Because eachchannel waveguide 18 of the arrayed-waveguide grating 16 has apredetermined waveguide length difference, each signal, after travelingover each channel waveguide 18 to the output slab waveguide 20, has apredetermined phase difference according to its waveguide lengthdifference. Since the phase difference depends on the wavelength of thesignal, each signal at different wavelength is focused on a differentposition along the arc boundary 28 of the output slab waveguide 20. As aresult, separated signals, each having a different wavelength, arereceived by the plurality of output channel waveguides 22 and emergetherefrom, respectively.

The general principles and performance of an AWGR multiplexer aresimilar to the AWGR demultiplexer, except that the direction ofpropagation of light is reversed, the ports that act as inputs for thedemultiplexer act as output ports for the multiplexer, and the portsthat act as output ports for the demultiplexer act as input ports forthe multiplexer.

Alternatively, an AWGR may comprise a plurality of output waveguides anda plurality of input waveguides; however, the general principles andperformance are similar to the AWGR demultiplexer.

Multiple routing functions including multiplexing and demultiplexing maybe integrated on a silicon wafer to form a complex planar lightwavecircuit (PLC). PLCs can be made using tools and techniques developed toextremely high levels by the semiconductor industry. Integratingmultiple components on a PLC may reduce the manufacturing, packaging,and assembly costs per function.

One aspect of performance that is affected by the present invention isreferred to as polarization dependent wavelength (PDW). This term, aswell as a number of related terms, will now be defined. Spectraltransmissivity (in units of dB) is defined as the optical power (inunits of dBm) of substantially monochromatic light that emerges from thefiber that is coupled to the input port minus the optical power (inunits of dBm) of the light that enters the optical fiber that is coupledto the output port of the optical router. Spectral transmissivity is afunction of the selected input port, the selected output port, theoptical wavelength, and the polarization state of the incident light.When the incident light is in a polarization state called a “principlestate of polarization,” the light will be in the same polarization statewhen it emerges from the device. For purposes of illustration only, theprinciple states of polarization are assumed to be independent ofwavelength, input port and output port. It is understood that theinvention is not so limited by this assumption. Again, for the purposesof illustration only, it will be assumed that the two principle statesof polarization are the so-called transverse electric (TE) andtransverse magnetic (TM) polarization states. The TE polarization statehas an electric field that is predominantly aligned in the transversedirection and the TM polarization state has an electric field that ispredominantly aligned in the lateral direction. Again, the invention isnot so limited to devices having these principle states of polarization.Typically, the device performance is sensitive to the polarization stateof the incident light is attributable to birefringence in the planarwaveguides and the channels waveguides comprising the AWGR.

FIG. 2A depicts, for a particular input/output port combination, a firstspectral transmissivity 40 associated with the TE polarization state anda second spectral transmissivity 42 associated with the TM polarizationstate. Typically, for values of spectral transmissivity that are largerthan −10 dB, the TM spectral transmissivity is a replica of the TEspectral transmissivity that is shifted in wavelength by an amount thatis referred to as the polarization dependent dispersion (PDD). HereinPDD is positive if the TM spectral transmissivity has a maximum that hasa longer wavelength than the maximum of the TE spectral transmissivityand is negative otherwise. Polarization dependent wavelength (PDW) isdefined herein as the absolute value of the PDD and is indicated in FIG.2A. The curves for the spectral transmissivity 46, 48, 50, 52 for fourinput/output combinations are shown together in FIG. 2C. The absolutevalue of the difference between the spectral transmissivities for TE andTM polarization states is referred to as the spectral polarizationdependent loss 44 and is depicted in FIG. 2B. The in-band PDL (IB-PDL)is the maximum value of the spectral polarization dependent loss withina specified wavelength range called a “band” (typically a 0.2 nm range)for a particular input port and output port. The PDL for the device istypically defined as the largest value of IB-PDL among the values ofIB-PDL for all input/output port combinations that are used in aparticular application. To meet typical application requirements, it iscritical for AWGRs to have a PDL value that is as close to 0 dB aspossible.

In typical fiber optic communication systems, the polarization state ofthe light in the optical fiber may change in a manner that isuncontrolled and unpredictable. A change in the polarization state ofthe light in the fiber as it enters an AWGR will cause a change in theoptical power that emerges from the AWGR that may be as large as thevalue of PDL for the AWGR. Because applications typically have littletolerance for such unpredictable changes in power, minimizing the PDL ofan AWGR is highly desirable. PDL can be minimized by minimizing PDW. Tomeet typical requirements, PDW may be required to be less than 0.05 nm.For this reason, the design and manufacture of an AWGR that has a lowvalue of PDW is highly desirable, yet very challenging.

There have been a number of techniques developed in an attempt tominimize PDW.

One approach to minimizing PDW involves selecting an optical layerdesign with minimum birefringence. In one example of this approach, U.S.Pat. No. 5,930,439 (Ojha et al) discloses a planar optical waveguidewhich reduces birefringence by doping the various optical layers so thatthe top cladding has a thermal coefficient of expansion that is close tothe thermal expansion coefficient of the substrate. This approach isappropriate for an optical layer design comprising deposited silicalayers with high concentrations of boron on a silicon substrate.Typically, this approach is impractical because the optical layers thatare required for low birefringence are not capable of surviving standardreliability tests. For example, the optical layers may absorb water andsubsequently form defects during a reliability test involving exposureto a temperature of 85° C. and a relative humidity of 85%,

A second approach requires the introduction of an optical waveplate. Forexample, U.S. Pat. No. 5,901,259 (Ando et al.) teaches forming anoptical waveplate by using a polyimide having a film thickness of 20 μmor smaller and further teaches the introduction of the waveplate onto anAWGR to reduce PDW. However, introducing a waveplate onto the AWGRtypically reduces the performance of the AWGR with respect to insertionloss, directivity, and return loss and occasionally may cause the AWGRto break. Furthermore, the introduction of a waveplate increases thecost associated with the production of the AWGR.

A third approach to reducing PDW involves waveguides of the AWG thatcomprise three segments, a central segment and two flanking segments. Afirst flanking segment has a birefringence equal to that to the secondflanking segment. The central segment has a birefringence that isdifferent from the flanking segments. The boundary around the centralwaveguide segments defines a region that is referred to herein as a“patch.” By selecting lengths of the segments that are appropriate tovalues of birefringence of the segments, an AWGR can be realized with asmall value for PDW. A variety of methods have been disclosed forproviding for segments with differing values of birefringence. Forexample, C. G. M. Vreeburg, et al. in “A low-loss 16-channelpolarization dispersion-compensated PHASAR demultiplexer,” IEEEPhotonics Technology Letters, Vol. 10, No. 3, Pp. 382-384 (1998)discloses a method wherein the AWG comprises InP-based rib waveguides,and the central segment differs from the flanking segments with respectto width of the rib and thickness of the top cladding region above therib. In general, waveguides may have birefringence contributions fromtwo independent sources, namely, form birefringence and stressbirefringence. For rib waveguides, changing the width of the waveguidechanges the form birefringence but does not substantially change thestress birefringence. For buried channel waveguide, changing the widthof the waveguide does not substantially change the form birefringence.The effect of the width of a buried channel waveguide on the value ofstress birefringence in the waveguide is not well known in the priorart.

In a second example, U.S. Pat. No. 5,341,444 (Henry et al.) discloses amethod that includes the deposition of a high index material, such assilicon nitride, in the patch region so that it is optically coupled tothe waveguide segments below it and thereby provides the centralsegments with a birefringence that is different from the birefringenceof the flanking segments.

In a third example, U.S. Pat. No. 5,623,571 (Chou et al.) discloses amethod that includes reducing the thickness of the cladding material inthe patch region so that waveguide segments below couple to the airabove the top cladding in the patch region and thereby provide thecentral segments with a birefringence that is different from thebirefringence of the flanking segments.

In a fourth example, C. K. Nadler et al. in “Polarization Insensitive,Low-Loss, Low-Crosstalk Wavelength Multiplexer Modules,” IEEE Journal ofSelected Topics in Quantum Electronics, Vol. 5, No. 5, pp. 1407-1412(1999), discloses a method for compensating polarization sensitivity ofAWGs by using “stress release” grooves etched on each side of thegrating waveguide in the central region.

In all of these examples of this approach, extra process steps arerequired to provide the waveguide segments within the patch region witha birefringence that is different from the flanking waveguide segments.The disclosed methods are difficult to implement in practice becauseproduction of the required optical layers within the patch region withinthe required tolerances is difficult. The added complexity associatedwith the production of two different optical layer designs in twodifferent regions also increases the cost of production. Despite theapproaches above, PDW remains a problem in current AWGR designs.

SUMMARY OF THE INVENTION

The present invention provides an arrayed waveguide grating withwaveguides of unequal widths. Described herein is an arrayed waveguidegrating where each waveguide of the grating preferably has asubstantially uniform width, but the width of any single waveguide inthe grating may be selected based on a predetermined birefringencepreferably required for the waveguide. Generally, the narrowest gratingwaveguide preferably has the longest overall length and the widestgrating waveguide preferably has the shortest overall length. Theremaining intermediate waveguides have widths that may be interpolatedbetween the narrowest and widest waveguide gratings. With an appropriatewidth for each waveguide, an arrayed waveguide grating may be providedhaving a low polarization dependent wavelength.

Alternative waveguides with variable widths may also be incorporated.For example, waveguides having two segments with different widths may beutilized. Alternatively, waveguides having tapered ends may also beutilized; and waveguides that taper in an arc-like pattern may also beutilized depending upon the desired results.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1A schematically depicts a conventional AWG with waveguides ofuniform width.

FIG. 1B depicts a cross-section from FIG. 1A showing the waveguidewidths.

FIG. 2A schematically depicts the spectral transmissivity for aparticular input/output port combination of an AWGR. The spectraltransmissivity for the TE and TM polarizations are illustrated. Thepolarization dependent wavelength (PDW) is indicated on the Fig.

FIG. 2B schematically depicts the spectral polarization dependent loss(PDL) and the in-band PDL.

FIG. 2C schematically depicts the spectral transmissivities for a singleinput port and four different output ports.

FIG. 3A schematically depicts a variation of an AWGR. In this variation,the width of each waveguide is independent of the distance along thewaveguide.

FIG. 3B depicts a cross-section from 3B-3B of FIG. 3A showing thegrating waveguides with unequal widths.

FIG. 4A schematically depicts a cross-section view of a buried channelwaveguide comprising parallel top and bottom surfaces, and side surfacesthat are not flat.

FIG. 4B schematically depicts a cross-section view of a buried channelwaveguide comprising parallel top and bottom surfaces, and flat sidesurfaces that are not parallel.

FIG. 4C schematically depicts a cross-section view of a buried channelwaveguide that has a substantially rectangular cross-section.

FIG. 5 schematically depicts a cross-section view of a rib waveguide.

FIG. 6 schematically depicts another variation of an AWGR. In thisvariation, the width of each waveguide varies only in the taper region.

FIG. 7A shows the distribution of PDD for a control group.

FIG. 7B shows the distribution of PDD for a first group.

FIG. 7C shows the distribution of PDL for the control group.

FIG. 7D shows the distribution of PDL for the first group.

FIG. 8A shows the spectral transmissivity of a first AWGR from thecontrol group.

FIG. 8B shows the spectral transmissivity of a second AWGR from thecontrol group.

FIG. 8C shows the spectral transmissivity of a first AWGR from the firstgroup.

FIG. 8D shows the spectral transmissivity of a second AWGR from thefirst group.

FIG. 9A shows the spectral transmissivity curves for several outputports of one AWGR of the control group.

FIG. 9B shows the spectral transmissivity curves for several outputports of one AWGR of the first group.

FIG. 10 schematically depicts another variation of an AWGR.

FIG. 11 schematically depicts another variation of an AWGR.

FIG. 12 schematically depicts another variation of an AWGR.

FIG. 13 schematically depicts another variation of an AWGR.

FIG. 14 schematically depicts another variation of an AWGR.

DETAILED DESCRIPTION OF THE INVENTION

Turning now to the drawings and referring initially to FIG. 3A, there isdepicted one variation of the inventive AWGR 60. AWGR 60 may comprise atleast one input waveguide 62, an input planar waveguide 64, an arrayedwaveguide grating (AWG) 66, an output planar waveguide 68, and at leastone output waveguide 70. An AWGR that operates as a demultiplexer mayrequire a plurality of output waveguides 70 and an AWGR that operates asa multiplexer may require a plurality of input waveguides 62. The AWGmay comprise a plurality of waveguides 66. An arbitrary number of curvedsurfaces, herein referred to as “normal surfaces” may be defined suchthat all waveguides of the AWG intersect the normal surfaces at normalincidence, i.e., such that the angle between the waveguide and thenormal surface is about 90°. Representative examples of normal surfacesare depicted as 78, 80, 82. A view of the AWGR in the normal surface isa cross-section view. In the present invention, the location where theAWG intersects at least one normal surface, at least one of thewaveguides of the AWG preferably has a birefringence value that isdifferent from the birefringence value from at least one other waveguideof the AWG. In this context, “birefringence value” refers to thebirefringence value of the fundamental guided mode of the waveguide. Thebirefringence values of the waveguides are significant because PDW ofthe AWG depends on the birefringence values of the waveguides.

For a typical channel waveguide, the birefringence value is a functionof the refractive indices of the materials that comprise the waveguideand the dimensions and shapes of various regions that comprise thewaveguide. As an example, detailed discussion is provided for thevariation in which the waveguides comprise three materials, namely thecore material, the bottom cladding material, and the top claddingmaterial, each with a refractive index that is uniform throughout thematerial. However, the invention is not so limited, and may apply, forexample, to waveguides that comprise any number of materials, each witha uniform refractive index, or alternatively, it may apply to gradedindex waveguides. It is preferred that the difference in birefringencevalues of different waveguides of the AWG may be attributable todifferences in the shape and dimensions of the core regions of thewaveguides. It is preferably that at least one of the waveguides of theAWG differs in cross-section from at least one other waveguide of theAWG.

For a typical buried channel waveguide, as illustrated in FIG. 4A, thecore 94 has a shape such that it comprises a bottom surface 96 that issubstantially flat, a top surface 98 that is substantially flat andsubstantially parallel to the bottom surface, and two side surfaces 100,102. The distance between the top surface of the core region and thebottom surface of the core region is referred to herein as the height,h, of the waveguide at a particular cross-section. When referring to aburied channel waveguide, the cross-sectional area of the core region ofthe waveguide divided by the height of the waveguide at a particularcross-section is referred to herein as the width of the waveguide.

FIG. 4B depicts a buried channel waveguide 104 comprising a core region106 with flat sides 108 and FIG. 4C, depicts a buried channel waveguide110 with a substantially rectangular cross-section. Although it ispreferred that the AWG waveguides comprise buried channel waveguides,the invention is not so limited and the AWG waveguides may compriseother waveguide types, for example, the AWG waveguides may comprise ribwaveguides 112, 112′, 112″ where each waveguide 112, 112′, 112″ may havea different corresponding height, h₁, h₂, h₃, respectively, as depictedin FIG. 5. In at least one cross-section view, at least one of thewaveguides of the AWG may preferably differ in width from at least oneother waveguide of the AWG. Providing the at least one waveguide with abirefringence value that is different from at least one other waveguideby providing the respective waveguides with appropriate widths ispreferred; however, it is also possible to produce waveguides havingdifferent heights or different refractive indices in accordance with theprinciples disclosed herein by, e.g., masking some but not all of thewaveguides of the grating such that each waveguide is exposed todifferent etch conditions or dopant diffusion conditions to providewaveguide cores of different heights, as shown in FIG. 5, or refractiveindex values respectively.

In one variation, at least one of the waveguides of the AWG has anaverage width that preferably differs from the average width of at leastone other waveguide of the AWG. In this context, “average width” refersto the average taken over a substantial portion of length of thewaveguide and is defined more clearly in the text and mathematicalexpressions that follow. As shown in FIG. 3A, the longest waveguide 76is preferably the narrowest relative to the other waveguides in grating66. It is further preferred for the core to be substantially rectangularin the cross-section view, as illustrated in FIG. 3B. A representativecross-section 71, section 3B-3B, of waveguide gratings 10 from FIG. 3Ais shown in FIG. 3B. There the waveguides are depicted with unequalwidths, as seen in first, intermediate, and N^(th) waveguides 72, 74,76, respectively. Also depicted are substrate 88, bottom cladding 90,and top cladding 92 in which waveguides 72, 74, 76 are disposed. Againas shown in this variation, the longest waveguide 76 is preferably thenarrowest relative to the other waveguides in grating 74 and 76. Andwaveguide 72 is preferably the widest waveguide. Intermediate waveguides74 may each have a width that progressively becomes narrower with eachadjacent waveguide from widest waveguide 72 to narrowest waveguide 76.This variation depicts AWGR 60 having four waveguides in grating 66, butthe number of waveguides here is merely exemplary and any number ofwaveguides may be utilized in practice. Likewise, any number ofwaveguides may be used such that the principles set forth herein may beapplied. AWG as shown in FIGS. 3A and 3B may be manufactured by anynumber of conventional methods as discussed below.

As generally set forth, a method of controlling birefringence in, e.g,an arrayed waveguide grating, in accordance with the present inventionmay comprise first transmitting a signal, e.g., a light signal, into theinput end of a first waveguide which may have a constant width and acertain length. The signal may also be transmitted into at least asecond waveguide which preferably has a width narrower than the width ofthe first waveguide and a length longer than the length of the firstwaveguide. As the number of waveguides increase, their correspondingwidths preferably decrease and corresponding lengths preferably increasein accordance with the principles discussed herein.

As discussed above, in making waveguide gratings of variable widths, anynumber of manufacturing methods may be utilized. Producing, e.g.,multiplexers and demultiplexers, may be integrated on, e.g., a siliconwafer to form a complex PLC. PLCs may be made using the tools andtechniques developed to extremely high levels by the semiconductorindustry.

In a typical embodiment, as shown in FIG. 3A, the slab waveguides 14 and20 may be comprised of at least three layers of doped silica depositedon a silicon wafer, namely, a bottom cladding layer that isapproximately 5 μto approximately 50 μm thick, a core layer that isapproximately 5 μm to approximately 12 μm thick and a top cladding layerthat is approximately 5 μm to approximately 50 μm thick. In a typicalvariation the core layer may have a refractive index that is larger thanthe refractive index of the bottom cladding between approximately 0.02and 0.2 and the refractive index of the bottom cladding is betweenapproximately 1.4 and 2.2 as measured in the wavelength range betweenabout 1520 nm and about 1600 nm. However, the invention is not solimited and may be applied to, e.g., integrated optical routerscomprising waveguides that are comprised of other materials such asInGaAsP, silicon, LiNbO₃, or polymer. Furthermore, the slab waveguidemay comprise more than three optical layers or may comprise a gradedindex layer. It is preferably that the input/output slab waveguidesand/or the input/output waveguides are single mode waveguides, i.e.,only the fundamental mode is guided by the waveguide.

The theory related the polarization dependence as it applies to aconventional AWGR and as it applies to the present invention ispresented below. The spectral transmissivity I depends on T, which isthe transfer function associated with the propagation of light from afirst curve, referred to herein at the “curve A” to a second curve,referred to herein as the “curve B.” At least some part of the AWG ispreferably between curve A and curve B. A representative example ofcurve A is shown as 82 in FIG. 3A and a representative example of curveB is shown as 78. In a second example, curve A 84 may be located at theinterface between the AWG 66 and the input planar waveguide 64, andcurve B may be located at the interface between the AWG 66 and theoutput planar waveguide 68. An expression for the transfer function isgiven by H. Yamada, K. Okamoto, A. Kaneko, and A. Sugita, DispersionResulting From Phase And Amplitude Errors In Arrayed-Waveguide GratingMultiplexers-Demultiplexers, Optics Letters, vol. 25, no. 8, pp. 569-571(2000). Accordingly, transfer function may be given by $\begin{matrix}{T = {\sum\limits_{i = 1}^{N}{a_{i}{\exp\left( {j\quad\phi_{i}} \right)}{\exp\left( {{jk}_{0}{OPL}_{i}} \right)}}}} & (1)\end{matrix}$where N is the number of waveguides in the AWG and OPL_(i) is theoptical path length along the ith waveguide of the AWG between curve Aand curve B; k₀ is the value of 2π/λ, where λ is the wavelength of thelight in vacuum that enters the input port of the AWGR; a_(i) and φ_(i)are real numbers that are determined in part on which input port andoutput port are being used and also determined in part by how lightpropagates from the input port of the AWGR to curve A. Variables a_(i)and φ_(i) may further be determined in part by how light would propagatefrom the output port to curve B if the direction of propagation of lightwere reversed with respect to the intended application. To simplifyfurther discussion, it may be assumed that curve A and curve B arecircular arcs. It may further be assumed that the input waveguidecouples light into the input planar waveguide at a location that iscentered on a point that is the center of curvature of curve A. It mayfurther be assumed that the output waveguide couples light from theoutput planar waveguide at a location that is centered on a point thatis the center of curvature of curve B. For an ideal Gaussian AWGR, φ_(i)has the same value for all waveguides of the AWG, i.e., φ_(i) isindependent of i. The relationship between I and T isI=TT*,  (2)where T* is the complex conjugate of T.

Generally in Eq. 1, a_(i), φ_(i) and OPL_(i) are dependent on thepolarization of light. To simplify further discussion, a_(i) may beassumed to be independent of polarization; however, the invention is notlimited to this assumption. Referring to the principle states ofpolarization as TE and TM, we may associate φ_(i) ^(TE), φ_(i) ^(TM),OPL_(i) ^(TE) and OPL_(i) ^(TM) with the values of φ_(i) and OPL_(i) forthe TE and TM polarization states, respectively. Eqs. 1 and 2 indicatethat I may have the same value for both polarizations of light (i.e. theAWGR will have zero PDW) ifOPL _(i) ^(TE)+φ_(i) ^(TE)=OPL_(i) ^(TM)+φ_(i) ^(TM)+δ  (3)where δ is an arbitrary constant. The value of δ has no effect on I; itmay be only important for δ to be independent of i. Equation 3 may beexpressed in the alternative formΔOPL _(i) ^(TE) =ΔOPL _(i) ^(TM)+Φ_(i)  (4a)whereΔOPL _(i) ^(TE) =OPL _(i+1) ^(TE) −OPL _(i) ^(TE)  (4b)ΔOPL _(i) ^(TM) =OPL _(i+1) ^(TM) −OPL _(i) ^(TM)  (4c)Φ_(i)=φ_(i+1) ^(TM)−φ_(i) ^(TM)−φ_(i+1) ^(TE)+φ_(i) ^(TE)  (4d)Eq. 4 is the criterion for zero PDW. It is preferred that, Φ_(i)=0;however, the invention is not so limited. The requirement expressed byEq. 4 depends only on the difference in OPL between adjacent waveguidesof the AWG. Consequently, the criterion for zero PDW (i.e., Eq. 4) maybe satisfied even if the evaluation of OPL values expressed in equation4 for any waveguide segment excludes a segment of each waveguide,provided that the each excluded segment has the same OPL. For example,the input tapers that couple light from the input planar waveguide maybe excluded from the evaluation of OPL for each waveguide withoutaffecting the interpretation of Eq. 4. In general OPL may be expressedby the equationOPL _(i) ^(TE) =n _(i) ^(TE) L ₁OPL _(i) ^(TM) =n _(i) ^(TM) L _(i)  (5)where n_(i) ^(TE) and n_(i) ^(TM) refer to the values of the effectiveindex of the fundamental mode of the waveguide averaged along thewaveguide between curve A and curve B for TE and TM polarized lightrespectively. Mathematically, n_(i) ^(TE) and n_(i) ^(TM) may beexpressed as $\begin{matrix}{{{n_{i}^{TE}L_{i}} = {\int_{0}^{L_{i}}{n_{z,i}^{TE}{\mathbb{d}z_{i}}}}}{{n_{i}^{TM}L_{i}} = {\int_{0}^{L_{i}}{n_{z,i}^{TM}{\mathbb{d}z_{i}}}}}} & (6)\end{matrix}$where n_(z,i) ^(TE) and n_(z,i) ^(TM) are the effective indices at adistance z along the waveguide from curve A towards curve B for the TEand TM polarizations respectively. Applying equations 5 and 6, equation4 may be expressed asL _(i) ΔB _(i) +B _(i) ΔL _(i)=Φ_(i)  (7)where ΔL_(i)=L_(i)+1−Li, ΔB_(i)=B_(i)+1−B_(i), andB _(i) =n _(i) ^(TM) −n _(i) ^(TE)  (8)or equivalently $\begin{matrix}{{B_{i}L_{i}} = {\int_{0}^{L_{i}}{B_{z,i}{\mathbb{d}z_{i}}}}} & (9)\end{matrix}$whereB _(z,i) =n _(z,i) ^(TM) −n _(z,i) ^(TE)  (10)In one variation of the inventive AWG, the AWG satisfies equation 7.

Eq. 7 does not necessarily fully specify the design of the AWG. Atypical additional constraining equation may be expressed asL_(i)Δn_(i)+n_(i)ΔL_(i)=Ψ₁ so that the pair of equations that define theAWG is given byL _(i) ΔB _(i) +B _(i) ΔL _(i)=Φ_(i)  (11a)L _(i) Δn _(i) +n _(i) ΔL _(i)=Φ_(i)  (11b)where 2n_(i)=n_(i) ^(TM)+n_(i) ^(TE), Δn_(i)=n_(i)+1−ni, and Ψ_(i) mayrepresent any number of expressions. Typically,Ψ_(i) =mλ ₀  (12)where m is the grating order of the AWGR is 10 and is an opticalwavelength that is typically the mean of the optical wavelengths thatare intended to be transmitted from one of the input ports to one of theoutput ports of the AWGR. An example of an alternative expression forΨ_(i) is $\begin{matrix}{\Psi_{i} = \left\{ \begin{matrix}{m\quad\lambda_{0}\quad\ldots\quad{{for}.{odd}.i}} \\{\left( {m + {1/2}} \right)\lambda_{0}\quad\ldots\quad{{for}.{even}.i}}\end{matrix} \right.} & (13)\end{matrix}$The above expression of Ψ_(i) provides for an AWGR with a passband thatis wider than the passband of a Gaussian AWGR. U.S. Pat. No. 5,467,418(C. Dragone) discloses other choices for Ψ_(i) that also provides for apassband that is wider than the passband of a Gaussian AWGR. In onevariation of the inventive AWG, the AWG satisfies equations 7 and 11. Inthe preferred variation, the values of B_(i) and n_(i) may be adjustedto the desired values by adjusting the values of w_(i) where w_(i) arethe values of the average widths of the waveguides, and where theaverage is taken along the length of the waveguides between curve A andcurve B. Mathematically the average width along a waveguide w_(i) andthe standard deviation of the width along a waveguide σ_(i) may bedefined herein as $\begin{matrix}{{{w_{i}L_{i}} = {\int_{0}^{L_{i}}{w_{z,i}{\mathbb{d}z_{i}}}}}{{\sigma_{i}^{2}L_{i}} = {\int_{0}^{L_{i}}{\left( {w_{z,i} - w_{i}} \right)^{2}{\mathbb{d}z_{i}}}}}} & (14)\end{matrix}$where w_(z,i) is the width of the ith waveguide at a distance z alongthe waveguide from curve A towards curve B. B_(i) and n_(i) may besubstantially varied by changing the width of the waveguide, which isnot obvious in the prior art. That B_(i) and n_(i) may be varied by therequired amount is discussed in further detail below.

For illustrative purposes only, the equations 7 and 11 are expressedbelow in an approximate manner such that the dependence of n_(i) andB_(i) is made explicit. $\begin{matrix}{{{{{L_{i}\left( \frac{\mathbb{d}B_{i}}{\mathbb{d}w} \right)}\Delta\quad w} + {B_{i}\Delta\quad L_{i}}} = \Phi_{i}}{{{{L_{i}\left( \frac{\mathbb{d}n_{i}}{\mathbb{d}w} \right)}\Delta\quad w} + {n_{i}\Delta\quad L_{i}}} = \Psi_{i}}} & (15)\end{matrix}$In Eq. 15 it is implicit that the derivatives are evaluated at w=w_(i).Eq. 15 is accurate to the extent that n_(i) and B_(i) have a lineardependence on w. Again for illustrative purposes only, Eq. 15 can beexpressed as $\begin{matrix}{{{{{L_{i}\left( \frac{\mathbb{d}B_{z,i}}{\mathbb{d}w_{z}} \right)}\Delta\quad w} + {B_{i}\Delta\quad L_{i}}} = \Phi_{i}}{{{{L_{i}\left( \frac{\mathbb{d}n_{z,i}}{\mathbb{d}w_{z}} \right)}\Delta\quad w} + {n_{i}\Delta\quad L_{i}}} = \Psi_{i}}} & (16)\end{matrix}$provided that${\left( \frac{\mathbb{d}B_{z,i}}{\mathbb{d}w_{z}} \right) = {{\left( \frac{\mathbb{d}B_{i}}{\mathbb{d}w} \right)\quad{and}\quad\left( \frac{\mathbb{d}n_{z,i}}{\mathbb{d}w_{z}} \right)} = \left( \frac{\mathbb{d}n_{i}}{\mathbb{d}w} \right)}},$i.e., provided that B_(z,i) and n_(z,i) are linearly depend on w_(z)within the range of values for w_(z) that are used in the waveguides. InEq. 16 it is implicit that the derivatives are evaluated atw_(z)=w_(z,i). The values of w_(i) and L_(i) for all the waveguides ofthe AWG can be determined from Eqs. 15 as follows. Values of Φ_(i) andΨ_(i) are assumed to be known as part of the design criteria. Values ofw₁ and L₁ are chosen. Then the values of n_(i), B_(i), and theirderivatives are evaluated for a width equal to w₁. Eqs. 15 are thenapplied to determine w₂ and L₂. The process is iterated to find all thevalues of w_(i) and L_(i) for i=2, . . . N. Alternatively, Eqs. 16 maybe applied in the same manner.

For illustrative purposes only, it is shown below that a closed formsolution for Δw_(i) and ΔL_(i) may be obtained provided that ΔL_(i) isindependent of i. $\begin{matrix}{{{- \left( {{n_{i}\frac{\mathbb{d}B_{i}}{\mathbb{d}w}} - {B_{i}\frac{\mathbb{d}n_{i}}{\mathbb{d}w}}} \right)}\Delta\quad L} = {{{\Phi_{i}\frac{\mathbb{d}n_{i}}{\mathbb{d}w}} - {\Psi_{i}\frac{\mathbb{d}B_{i}}{\mathbb{d}w}} - {{L_{i}\left( {{n_{i}\frac{\mathbb{d}B_{i}}{\mathbb{d}w}} - {B_{i}\frac{\mathbb{d}n_{i}}{\mathbb{d}w}}} \right)}\Delta\quad w_{i}}} = {{B_{i}\Psi_{i}} - {n_{i}\Phi_{i}}}}} & (17)\end{matrix}$The subscript on ΔL_(i) has been omitted to indicate that ΔL_(i) doesnot have a dependence on i in Eq. 17. As a typical example, it isfurther assumed that Φ_(i)=0 and Ψ_(i)=mλ₀, then Eq. 17 may be expressedas $\begin{matrix}{{{- \left( {{n_{i}\frac{\mathbb{d}B_{i}}{\mathbb{d}w}} - {B_{i}\frac{\mathbb{d}n_{i}}{\mathbb{d}w}}} \right)}\Delta\quad L} = {{{{- m}\quad\lambda_{0}\frac{\mathbb{d}B_{i}}{\mathbb{d}w}} - {{L_{i}\left( {{n_{i}\frac{\mathbb{d}B_{i}}{\mathbb{d}w}} - {B_{i}\frac{\mathbb{d}n_{i}}{\mathbb{d}w}}} \right)}\Delta\quad w_{i}}} = {m\quad\lambda_{0}B_{i}}}} & (18)\end{matrix}$One result that follows from Eq. 18 is(dB _(i) /dw)Δw_(i) =−B _(i) ΔL/L _(i)  (19)Typically, (dB_(i)/dw)/B_(i) is positive; hence it is preferred toarrange the average widths of the waveguides of the AWG such that theshortest waveguide is, on average, the widest and that the longestwaveguide is, on average, the narrowest. Since B_(i) varies slightly asw_(i) changes, Eq. 19 suggests that variation for which ΔL_(i) isconstant and Φ_(i)=0 and Ψ_(i)=mλ₀ it is preferred to have waveguideswith widths w_(i) that vary according to the number of the waveguide iin a manner that is slightly nonlinear.

The average widths of the waveguides of the AWG can be determined fromeither Eqs. 11, Eqs. 15, Eqs. 16, Eqs. 17, or Eqs. 18. For anyparticular waveguide that is required to have a particular averagewidth, a number of variations are possible.

First Example

In one example, which is illustrated in FIGS. 3A and 3B, the width ofthe waveguide may have a constant value throughout the entire length ofthe waveguide between the two planar waveguides, i.e., the width of thewaveguide for any cross-section has a value that is equal to therequired average value. In this variation, the standard deviation inwidth along each waveguide is substantially equal to zero. Typically,the values of optical path length that are used in the designcalculations are based on the optical path lengths curve A 84 located atthe juncture between the AWG region 66 and the input planar waveguide 64and curve B 86 located at the juncture between the AWG region 66 and theoutput planar waveguide 68.

Second Example

In a second example, which is illustrated in FIG. 6, each waveguide ofthe AWG preferably has a constant width throughout the length of thewaveguide except for a taper region where the AWG waveguide couples tothe input planar waveguide (herein referred to as the “AWG input taper”)and for a taper region where the AWG waveguide couples to the outputplanar waveguide (herein referred to as the “AWG output taper”). Thevariation 120 is similar to that shown in FIG. 3A in that grating 122depicts first, intermediate, and Nth waveguides 124, 126, 128,respectively. Also, a cross-sectional view 3B-3B of waveguides 122 isshown similar to FIG. 3B. However, in FIG. 6, the AWG input taper isbetween the input planar waveguide 64 and curve A 82, and the AWG outputtaper is between the output planar waveguide 68 and curve B 78. In thisvariation, it is preferred that each AWG input taper have a length thatis substantially equal to the length of all other AWG input tapers andthat each AWG output taper have a length that is substantially equal tothe length of all other AWG output tapers. In this variation, it isfurther preferred that each AWG input taper have values of effectiveindex=n_(z,i) ^(TM)−n_(z,i) ^(TE) that vary linearly throughout thetaper region according to the mathematical functionsn _(z,i) ^(TE) =n _(i) ^(TE)+(n ₁ ^(TE) −n _(i))z/L ₁  (20a)n _(z,i) ^(TM) =n _(i) ^(TM)+(n ₁ ^(TM) −n _(i))z/L ₁  (20b)where z is the distance along the waveguide from curve A towards theinput planar waveguide and that each AWG output taper have values ofeffective index n_(z,i) ^(TM) and n_(z,i) ^(TE) that vary linearlythroughout the taper region according to the mathematical functionsn _(z,i) ^(TE) =n _(i) ^(TE)+(n _(o) ^(TE) −n _(i))z/Lo  (21a)n _(z,i) ^(TM) =n _(i) ^(TM)+(n _(o) ^(TM) −n _(i))z/L _(o)  (21b)where z is the distance along the waveguide from curve A towards theinput planar waveguide. Provided that n_(z,i) ^(TM) and n_(z,i) ^(TE)vary linearly with the width of the waveguide, Eqs. 20 and 21 aresatisfied by a linear taper design. If Eq. 20 is satisfied, then OPL_(i)^(TE) and OPL_(i) ^(TM) may be given byOPL _(i) ^(TE)=(L ₁ n ₁ ^(TE) +L _(o) n _(o) ^(TE))/2+L _(i) n _(i)^(TE)  (22a)OPL _(i) ^(TM)=(L ₁ n ₁ ^(TM) +L _(o) n _(o) ^(TM))/2+L _(i) n _(i)^(TM)  (22b)where n_(i) ^(TM) and n_(i) ^(TM) now refers to the effective indexvalues of the waveguide averaged over the section of the waveguide thatexcludes the tapers and Li refers to the length along the waveguide fromthe midpoint of the input taper to the midpoint of the output taper.Applying Eq. 22, yields the resultL _(i) Δn _(i) +n _(i) ΔL _(i)=Ψ_(i)  (23a)which is the same as Eq. 11b, except that n_(i) now refers to aneffective index value of the waveguide averaged over the section of thewaveguide that excludes the tapers. Similarly, the following result maybe obtainedL _(i) ΔB _(i) +B _(i) ΔL _(i)=Φ_(i)  (23b)which is similar to Eq 11a, except that in Eq. 23a, Bi refers to thevalue of birefringence averaged over the section of the waveguide thatexcludes the tapers. Since Eq. 23 is formally equivalent to Eq. 11, theequations above that are derived from Eq. 11 may apply to this variationprovided the modified interpretation of L_(i), n_(i) i and B_(i) areused.

Experimental Results of the Second Example

To demonstrate the utility of the present invention, a first group ofAWGRs were fabricated that were based on the variation of FIG. 6 and asecond group of additional conventional AWGRs were fabricated (referredto herein as the control group). The control group was similar in designto the first group except that all waveguides of the AWG had the samewidth. Three input waveguides were tested for each AWGR of each group.In both groups, 197 waveguides were used in the AWG. In the first group,the range of waveguide widths, i.e., value of |w_(N)−w₁| was between 1μm and 3 μm and the value of w_(N)+w₁ was between 15 μm and 18 μm. Inthe control group, the value of |w_(N)−w₁| was less than 0.4 μm and thevalue of w_(N)+w₁ was between 15 μm and 18 μm. Three inputs were testedfor each AWGR. At least 40 outputs were tested for each input waveguideof each AWGR. For the control group, the average value of PDD was 0.10mm and the standard deviation of the values of PDD was 0.02 nm.

The distribution of the measured values of PDD and PDL for the controlgroup is shown in FIGS. 7A and 7C, respectively. For the control group,the average value of PDL was 0.58 dB and the standard deviation of thePDL values was 0.14 dB. For the first group, the average value of PDDwas 0.00 nm and the standard deviation of the values of PDD was 0.02 nm.For the control group, the average value of PDL was 0.15 dB and thestandard deviation of the PDL values was 0.05 dB.

The distribution of the measured values of PDD and PDL for the firstgroup is shown in FIGS. 7B and 7D respectively. Applying a typicaldevice requirement of 0.05 nm for the maximum value of PDD, the controlgroup on average fails to meet the required value of PDW and the firstgroup successfully meets the required value of PDW. Furthermore, theapplication of the invention does not substantially degrade the noisefloor of the spectral transmissivity. This may be evident by thecomparison of FIG. 8A against FIG. 8C and by the comparison of FIG. 8Bagainst FIG. 8D. FIGS. 8A and 8B depict the measured spectraltransmissivities for two AWGRs of the control group and FIGS. 8C and 8Dshow two AWGRs of the first group. Further comparison between an AWGR ofthe first group and an AWGR of the control group may be provided byFIGS. 9A and 9B. FIG. 9A depicts the spectral transmissivity curves ofseveral adjacent output waveguides of one AWGR of the control group.FIG. 9B depicts the spectral transmissivity curves of several adjacentoutput waveguides of one AWGR of the first group. The PDW of the AWGRfrom the first group is shown to be smaller than the PDW of the AWGR ofthe control group. Also, the passband shape is not substantially changedby the application of the invention.

Third Example

In a third example, illustrated in FIG. 10, each waveguide of the AWGmay comprise about seven segments 130, 132, 134, 136, 138, 140, 142.Segments 130, 134, 138, 142 are taper segments in which the width of thewaveguide preferably changes and segments 132, 136, 140 are segments inwhich the width of the waveguide preferably does not change. Segments130 and 142 are located in a region of the AWG where the opticalcoupling between adjacent waveguide may be very strong. Segments 134,136, 138 may be located in a region of the AWG where the opticalcoupling between adjacent waveguides may be very weak. The width at eachend of segment 130 and the length of segment 130 is preferably the samefor all waveguides, consequently, the OPL of segment 130 may be the samefor all waveguides. Similarly, the width at each end of segment 142 andthe length of segment 142 is preferably the same for all waveguides,consequently, the OPL of segment 142 may be the same for all waveguides.Similarly, the width and length of segment 132 is preferably the samefor all waveguides, consequently, the OPL of segment 132 may be the samefor all waveguides. Similarly, the width and length of segment 140 ispreferably the same for all waveguides, consequently, the OPL of segment140 may be the same for all waveguides. Consequently, segments 130, 132,140, 142 may be excluded from the evaluation of the OPLs that enter intoEq. 4 and all equations derived from Eq. 4 without affecting thevalidity of Eq. 4 as the criterion for zero PDW. This variation ispreferable if the evaluation of the input and output tapers of thesecond embodiment have OPLs that are different from one another and ifthe coupling between segments makes it difficult to evaluate the valuesof OPL in this region.

Fourth Example

In a fourth example, which is illustrated in FIG. 11, the ith waveguidemay comprise two segments, a first segment 144 of width w_(A) and lengthL_(Ai) and a second segment 146 of width w_(B) and length L_(B,i).Because each segment may have a different width, each segment mayconsequently have a different effective index and birefringence. Theaverage width, average effective index and average birefringence may begiven byw _(i) L _(i) =w _(A) L _(A,i) +w _(B) L _(B,i)n _(i) L _(i) =n _(A) L _(A,i) +n _(B) L _(B,i)B _(i) L _(i) =B _(A) L _(A,i) +B _(B) L _(B,i)the average values of width, effective index and birefringence, in thisvariation preferably depend on i since L_(A,i) and L_(B,i) depend on i.Using the average values expressed above in Eq. 11 yields the followingresultB _(A) ΔL _(A,i) +B _(B) ΔL _(B,i)=Φ_(i)  (24a)n _(A) ΔL _(A,i) +n _(B) ΔL _(B,i)=Ψ_(i)  (24b)where n_(A) and n_(B) are the effective index values of first and secondsegments 144, 146, respectively, and B_(A) and B_(B) are thebirefringence values for first and second segments 144, 146,respectively. All the values of L_(A,i) and L_(B,i) may be found in thefollowing manner. L_(A,1) and L_(B,1) are chosen. The values of n_(A),n_(B), B_(A), and B_(B) may be determined from the waveguide structureof the respective segment. Eq. 24 may be applied to determine L_(A,2)and L_(B,2). Eq. 24 may then be applied iteratively until L_(A,i) andL_(B,i) values are found for i=2, . . . N.

Fifth Example

A fifth example is illustrated in FIG. 12. This example is similar tothe third example, except that tapers are used on each waveguide betweenfirst segment 148 and second segment 150. In this example, the taperdesign may be the same on all waveguides; therefore, the taper betweensegment A and segment B of each waveguide may be excluded from theevaluation of the OPL of each waveguide when applying Eq. 4 or any ofthe equations derived from it.

Sixth Example

A sixth example is illustrated in FIG. 13. In this example, the ithwaveguide may comprise three segments, a central segment 154 and twoflanking segments 152, 152′. A first flanking segment 152 preferably hasa birefringence equal to that to the second flanking segment 152′. Thecentral segment 154 may have a birefringence that is different from theflanking segments 152, 152′. The boundary around the central waveguidesegments 156 preferably defines a region that is referred to herein as a“patch.” Such patches may typically be placed symmetrically about amidpoint of the relevant waveguides although the patches mayindividually be positioned anywhere along the waveguides. By selectinglengths of the segments that are appropriate to values of birefringenceof the segments, an AWGR may be realized with a small value for PDW. Thethree segments 152, 154, 152′ may have widths w_(A), w_(B), and w_(A),respectively, i.e., segments 152 and 152′ may have equal widths. Eq. 24may be applied to this example provided that L_(Ai) is interpreted andthe length of segment 152 plus the length of segment 152′. The number ofwaveguides and widths depicted here is merely exemplary and is not meantto limit the scope of this invention. Any number of waveguides, widths,and any number of patches having variable widths may be utilizeddepending upon the desired application and remains within the scope ofthis invention.

Seventh Example

A seventh example is illustrated in FIG. 14. In this example, eachwaveguide of the AWG preferably has a width that continuously changes inan arc-like manner such that the waveguide may have a certain width atsegment 158. The width then preferably tapers in a gentle arc to athinner segment 160; and the waveguide then tapers gently to a width atsegment 162, preferably similar to segment 158.

Although the present invention has been described with reference toparticular variations, the description is only an example of theinvention's application and should not be taken as a limitation. Variousadaptations and combinations of features of the variations disclosed arenow readily apparent to those of ordinary skill and are within the scopeof the invention.

1-32. (canceled)
 33. A method of providing a predetermined polarizationdependent wavelength in an optical device comprising: transmitting lightinto a first waveguide having a first width and a length; transmittingthe light into at least a second waveguide having a second narrowerwidth and a longer length.
 34. The method of claim 33 further comprisingtransmitting the light into a plurality of additional waveguides,wherein each of the additional subsequent waveguides has an increasinglength and a corresponding decreasing width.
 35. The method of claim 33wherein the light is transmitted into the first and the secondwaveguides simultaneously.
 36. The method of claim 33 wherein the lightis transmitted into the first waveguide before the second waveguide. 37.The method of claim 33 wherein the light is transmitted into the secondwaveguide before the first waveguide.